I'll give you the simple version.
The oxygen sensor is screwed into the exhaust, where it monitors the presence or abscence of oxygen.
Based on the information from this sensor, yhe vehicles computer will either richen or lean the air/fuel mixture to the engine.
A failed sensor will illuminate a " check engine " light, your fuel economy will be decreased, accompanied by an increase of exhaust emissions.
It has absolutely no effect on the startability of an engine. ;)
Btw., if you have a diesel, forget everything I said. :D